Advanced Calculus Single Variable

10.3 The Young Integral

This dates from about 1936. Basically, you can do ∫0TY dF if Y is continuous and F is of
bounded variation or the other way around. This is the old Stieltjes integral. However, this
integral has to do with Y ∈ Vq and F ∈ Vp and of course, these functions are not necessarily
of bounded variation although they are continuous. First, here is a simple lemma which is
used a little later.

Lemma 10.3.1Let F be piecewise linear on a dissection P and let Y be continuous.Then